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Appl. Appl. Math.
ISSN: 1932-9466
Applications and Applied
Mathematics:
An International Journal
(AAM)
Vol. 8, Issue 2 (December 2013), pp. 523 – 534
A New Approach to the Numerical Solution of
Fractional Order Optimal Control Problems
T. Akbarian and M. Keyanpour
Department of Applied Mathematics
Faculty of Mathematical Sciences
University of Guilan
Rasht, Iran
Akbarian.t@gmail.com; m.keyanpour@gmail.com
Received: July 24, 2012; Accepted: September 9, 2013
Abstract
In this article, a new numerical method is proposed for solving a class of fractional order optimal
control problems. The fractional derivative is considered in the Caputo sense. This approach is
based on a combination of the perturbation homotopy and parameterization methods. The control
function u(t) is approximated by polynomial functions with unknown coefficients. This method
converts the fractional order optimal control problem to an optimization problem. Numerical
results are included to demonstrate the validity and applicability of the method.
Keywords: Fractional order optimal control, Homotopy perturbation method, Caputo
fractional derivative.
AMS-MSC 2010 No.: 49J21, 26A33
1. Introduction
Fractional optimal control problems (FOCPs) are optimal control problems associated with
fractional dynamic systems. The fractional optimal control theory is a very new topic in
mathematics. FOCPs may be defined in terms of different types of fractional derivatives. But the
most important types of fractional derivatives are the Riemann-Liouville and the Caputo
fractional derivatives. In Agrawal (2004), Agrawal and Baleanu (2007) the authors obtained
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T. Akbarian and M. Keyanpour
necessary conditions for FOCPs with the Riemann-Liouville derivative and were able to solve
the problem numerically. Agrawal (2008a) presented a quadratic numerical scheme for a class
of fractional optimal control problems (FOCPs). In Agrawal (2008c), the FOCPs are formulated
for a class of distributed systems where the fractional derivative is defined in the Caputo sense,
and a numerical technique for FOCPs presented. Baleanu et al. (2009) used a direct numerical
scheme to find a solution of the FOCPs. In Biswas and Sen (2011a), FOCPs with fixed final
time are considered and a transversely condition is obtained. FOCPs with dynamic constraint
involving integer and fractional derivatives are also considered Biswas and Sen (2011b). Based
on the expansion formula for fractional derivatives, a new solution scheme was proposed in
Jelicic and Petrovacki (2009). Lotfi et al. (2011) used Legendre orthonormal polynomial basis to
solve the FOCPs.
A direct method using Eigen functions to solve the FOCPs of a 2-dimensional system was
presented in ܱሷzdemir et al. (2009), where the Gr‫ݑ‬ሷ nwald-Letnikov approximation was used to
approximate the fractional derivatives. Similar attempts have been made by several researchers
for solving the FOCP of distributed systems (Hasan et al. (2011), Rapaic and Jelicic (2010)).
Tricaud and Chen (2010a) presented a numerical scheme for FOCPs based on integer order
optimal controls problem.
In Youse et al. (2011) the usage of Legendre multiwavelet basis and collocation method was
proposed for obtaining the approximate solution of FOCPs. Tricaud and Chen (2010b) proposed
a rational approximation based on the Hankel data matrix of the impulse response to obtain a
solution for the general time-optimal problem. The interested reader is referred to Evirgen and
ܱሷzdemir (2012), Wang and Zhou (2011), Jarad (2010), ܱሷzdemir (2009), Agrawal (2008b), and
Frederico et al. (2008) for further information.
The homotopy perturbation method (HPM) was applied to solve ODE and PDE equations in He
(2003), He (2005), Biazar and Ghazvini (2009). The numerical method for FOCPs presented in
this paper follows the approach presented in Keyanpour and Azizsefat (2011), Keyanpour and
Akbarian (2011), Borzabadi et al. (2010), Borzabadi and Azizsefat (2010). Of course in this
paper we develop a hybrid of parametrization and modified homotopy perturbation method to
solve FOCPs.
This paper is organized as follows: In Section 2 we present some basic definitions. In section 3
we describe our method. In section 4 we report our numerical results. Finally section 5 we
conclude the paper.
2. Fractional Optimal Control Problem Statement
2.1. Basic Definitions
We present some basic definitions related to fractional derivatives. The Left Riemann-Liouville
derivative of fractional order m  1    m for a function f (t ) is defined by:
AAM: Intern. J., Vol. 8, Issue 2 (December 2013)
Dt f ( x) 
a
x
1
d
( ) m  ( x  t ) m 1 f (t )dt ,
(m  ) dx a
525
(1)
while the Right Riemann-Liouville fractional derivative is given as:
t
Db f ( x ) 
b
1
d
(  ) m  (t  x) m  1 f (t )dt.
(m   ) dx x
(2)
Another fractional derivative, the left Caputo fractional derivative, is defined as:

a D*t f ( x ) 
m
x
1
m  1 d f (t )
(
t

x
)
(
)dt ,
(m  ) a
dt m
(3)
while the right Caputo fractional derivative given by:

t D*b f ( x ) 
m
b
1
f (t )
m  1
m d
(
t

x
)
(

1
)
(
)dt ,

(m   ) x
dt m
(4)
where m   . 2.2. Fractional Optimal Control Problem Formulation
The FOCPs in the sense of Caputo are formulated as follows:
b
Minimize J ( x, u )   f (t , x (t ), u (t ))dt ,
(5)
a
subject to:
a
D*t x(t )  G ( x(t ), u (t ), t ),
(6)
the initial conditions of the problems are (k = m-1).
x k (a )  c k ,
x k (b)  s k ,
where
(7)
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T. Akbarian and M. Keyanpour
G (t )  ( g1 (t ), , g l (t )) t , u (t )  (u1 (t ),  , u m (t )) t ,
x(t )  ( x1 (t ),  , xl (t )) t ,


f  C a, b R l  R m ,
in which x and u are the state and control variables respectively, t  a, b stands for the time and
f and G are given nonlinear functions. Here, we assume that FOCPs have a unique solution.
The basic existence and uniqueness follow from the Lipschitz condition by using contraction
mapping theorem and a weighted norm with Mittag-Leffler in Yakar et al. (2012),
Lakshmikantham and Mohapatra (2001), Podlubny (1999), Samko et al. (1993), Shaw and Yakar
(1999).
3.
Description of Method
In this section the proposed method is described and an associated algorithm is presented. The
continuous control function u (t ) is approximated with a finite combination of elements of a
basis Rudin (1976) as follow:
r
u (t )   w j q j .
(8)
j 0
Since the FOCPs are solved by homotopy perturbation method,
homotopy as follows:

 dx m
dx m
 m  G  x(t ), u (t ), t  a D*t x(t ) , p  0,1

p
m
dt

 dt
we construct a convex
(9)
and suppose the solution of Equation(6) has the following form:
(10)
x  x0  px1  p 2 x 2  ,
where x j (t , w0 , w1 ,, wr ), j  0,1,, r are unknown functions. Substituting Equation (10) into
Equation (9) for x (t ) , and equating the coefficients of the terms with identical powers of p , we
derive:
 0 dx0m
x0k (a )  ck ,
p : m 0
dt

 1 dx1m dx0m

 p : m  m  G ( x0 (t ), u (t ), t )  a D*t x0 (t ),
dt
dt


dx m dx m
 p 2 : 2m  1m  G ( x1 (t ), u (t ), t )  a D*t x1 (t ),
dt
dt

 .
x1k (a )  0,
x2k (a )  0,
(11)
AAM: Intern. J., Vol. 8, Issue 2 (December 2013)
527
As p  1, Equation (9) tends to Equation (6) and Equation (10), in most cases converges to an
approximate solution of Equation (6), i.e.,
(12)
x  x0  x1  x2 .
By substitution of Equation (8) and Equation (12) into Equation (5) and Equation (7) we obtain
an approximate solution of FOCPs as follows:
b
r
a
j 0
( w0 ,, wr )
J r ( w0 ,, wr )   f (t , x(t , w0 ,, wr ),  w j q j (t ))dt ,
s.t :
x k (a, w0 , w1 ,, wr )  c k ,
min
(13)
x k (b, w0 , w1 ,, wr )  s k .
Let J r* be the optimal value of Equation (13). A stopping criterion is proposed as follows:
J r*  J r*1   ,
(14)
where the small positive number  is chosen according to the accuracy desired. We propose the
following algorithm, which is presented in two stages.
Algorithm:
Initialization step:
Choose  for the accuracy desired and set r  1 , and go to the main step.
Main step
Step 1. Set u (t ) by Equation (8) and go to Step 2.
Step 2. Compute x (t ) by Equation (12) and go to Step 3.
Step 3. Then compute J r*  inf J r by Equation (13). If r  1 go to step (5). Otherwise, go to
Step 4.
Step 4. If the stopping criterion Equation (14) holds, then stop; else , go to Step 5.
Step 5. r  r  1 and go step 1.
3. Numerical Results
In this section we apply the method presented in Section 3 to solve the following two test
examples. All computations carried out by the package MAPLE 13.
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T. Akbarian and M. Keyanpour
Example 1. Consider the following time invariant problem
J


1 1 2
x (t )  u 2 (t ) dt , 
0
2
(15)
subject to:
0
D*t x(t )   x (t )  u (t ),
with the initial condition
x(0)  1.
The exact solution for   1 is
 
 2t , u (t )  1  2  cosh  2t    2   sinh  2t , x (t )  cosh 2t   sinh
where
 
 
 2   0.98,
2 cosh  2   sinh  2 
cosh 2  2 sinh
with objective value J * ( x, u )  0.192909. In Figure 1, the state variable x (t ) and the control
variable u (t ) are plotted for   1. It is obvious that by applying the algorithm presented in
section 3, the approximate values of x (t ) and u (t ) converge to the exact solutions. Figure 2
shows the state x (t ) and the control input u (t ) as functions of time t for different values of  .
Choosing   10 5 , the results of the applying the given algorithm are presented in Table 1.
Table 1. Numerical results in Example 1

n J* r
1
4
7 0.192909
0.99
6
6 0.19153
0.9
7
6 0.17952
0.8
6
5 0.16729
AAM: Intern. J., Vol. 8, Issue 2 (December 2013)
529
Figure 1. (a) Approximate solutions and exact solution of u (t ) for (n  7,  1)
(b) Approximate solutions and exact solution of x (t ) for ( n  7,  1) for Example 1
(a)
(b)
Figure.2: (a) State x (t ) as a function of t for different values of 
(b) Control u (t ) as a function of t for different values of 
for Example 1
Example 2. In this example, a time varying FOCP is considered to find the control u (t ) , which
minimizes the performance index
530
T. Akbarian and M. Keyanpour
J


1 1 2
x (t )  u 2 (t ) dt ,

0
2
(16)
subject to:
0
D*t x(t )  tx (t )  u (t ),
with free terminal condition and the initial condition
x(0)  1.
Figure 3 demonstrates the approximation of x (t ) and u (t ) for different values of  . The results
of applying the algorithm are presented in Table 2. Figure 4 shows the state and the control
variables, respectively, as a function of time for   0.8 for different values of n . It is obvious
that the approximate values x (t ) and u (t ) converge to the exact solutions by increasing the
values of n .
Table 2. Numerical results in Example 2

n J* r
1
3
5 0.48427
0.99
3
5 0.48347
0.9
3
5 0.47605
0.8
3
5 0.46722
(a)
(b)
Figure 3. (a) State x (t ) as a function of t for different values of 
(b) Control u (t ) as a function of t for different values of 
for Example 2
AAM: Intern. J., Vol. 8, Issue 2 (December 2013)
531
(a)
(b)
Figure 2: (a) Convergence of the state variable for the time-varying system for   0.8
(b) Convergence of the control variable for the time- varying system for   0.8.
Test problems 1 and 2 were solved in Agrawal (2008a) in a different way. Our results, shown in
Figures 1-4 are in good agreement with the results demonstrated in Agrawal (2008a). But, we
achieved satisfactory numerical results in only 5 iterations while in Agrawal (2008a), the number
of approximations starts in 10 and increases up to 320. So it is significant that we achieved our
numerical results with a very small order of approximations. Also we find the approximate
optimal value of the objective function for each .
4. Conclusion
In this paper, we have developed the homotopy perturbation and parameterization methods for
solving a class of fractional optimal control problems. By the proposed method we are able to
reduce the main problem to an optimization problem. The numerical results have demonstrated
the high accuracy of the proposed method.
Acknowledgments
We are very grateful to two anonymous referees for their careful reading and valuable comments
which led to the improvement of this paper.
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T. Akbarian and M. Keyanpour
REFERENCES
Agrawal, Om Prakash (2008a). A quadratic numerical scheme for fractional optimal control
problems. ASME Journal of Dynamic Systems, Measurement, and Control, Vol. 130, No. 1,
011010-1-011010-6.
Agrawal, Om Prakash (2008b). A formulation and numerical scheme for fractional optimal
control problems. J. Vib. Control. Vol. 14, 1291-1299.
Agrawal, Om Prakash (2004). A general formulation and solution scheme for fractional optimal
control problems, Nonlinear Dynamics, Vol. 38, No. 14, 323-337.
Agrawal, Om Prakash and Baleanu, Dumitru (2007). A Hamiltonian formulation and a direct
numerical scheme for fractional optimal control problems, Journal of Vibration and Control,
Vol. 13, No. 9-10, 1269-1281.
Agrawal, Om Prakash (2008c). Fractional optimal control of a distributed system using
eigenfunctions, ASME Journal of Computational and Nonlinear Dynamics, Vol. 3, No. 2,
021204-1-021204-6.
Borzabadi, Akbar Hashemi, Azizsefat, Mojtaba, and Fard, Omid Solimani (2010). Homotopy
perturbation method for optimal control problems governed by Volterra integral equations,
JARDCS, Vol. 2, No. 2, 13-25.
Baleanu, Dumitru, Defterli, Ozlem, and Agrawal, Om Prakash (2009). A Central Difference
Numerical Scheme for Fractional Optimal Control Problems, Journal of Vibration and
Control, Vol. 15, No. 4, 583-597.
Biazar, Jafar, and Ghazvini, Hosein (2009). Convergence of the homotopy perturbation method
for partial differential equations, Nonlinear Analysis,Real Word Applications, Vol. 10 ,
2633--2640.
Borzabadi, Akbar Hashemi, and Azizsefat, Mojtaba (2010). Numerical solution of optimal
control problems governed by integro-differential equations via a hybrid iterative scheme,
WASJ, Vol. 10, IssN 1818-4952, 538-543.
Biswas, Raj Kumar, and Sen, Siddhartha (2011a). Fractional optimal control problems: a
pseudo-state-space approach. Journal of Vibration and Control, Vol. 17, No.7, 1034–1041.
Biswas, Raj Kumar, and Sen, Siddhartha (2011b). Fractional optimal control problems with
specified final time. ASME Journal of Computational and Nonlinear Dynamics, Vol. 6,
021009.1–021009.6.
Evirgen, Firat, and ܱሷzdemir, Necati (2012). A Fractional Order Dynamical Trajectory Approach
for Optimization Problem with HPM. Springer, Eds. Baleanu, D., Machado, J.A.T., Luo,
A.C.J., Fractional Dynamics and Control, ISBN 978-1-4614-0456-9, 145-155.
Frederico, Gastao S. F. and Torres, Delfim F. M. (2008). Fractional conservation laws in optimal
control theory, Nonlinear Dynamics, Vol. 53, No.3, 215-222.
He, Ji-Huan (2003). Homotopy perturbation method, A new nonlinear analytical technique,
Appl.Mathe. Compu, Vol. 135, 73-79.
He, Ji-Huan (2005). Application of homotopy perturbation method to nonlinear wave equations,
Chaos Soliton. Fract, Vol. 26, 695-700.
Hasan, M. Mehedi, Tangpong, Xiangqing W, and Agrawal, Om Prakash (2011). Fractional
optimal control of distributed systems in spherical and cylindrical coordinates. J. Vib.
Control, doi:10.1177/1077546311408471.
Jarad, Fahd, Abdeljawad, Thabet and Baleanu Dumitru (2010). Fractional variational optimal
control problems with delayed arguments. Nonlinear Dynamics, Vol. 62, No. 3, 609–614.
AAM: Intern. J., Vol. 8, Issue 2 (December 2013)
533
Jelicic, Zoran. D, and Petrovacki, Nebojsa (2009). Optimality conditions and a solution scheme
for fractional optimal control problems, Structural and Multidisciplinary Optimization, Vol.
38, No. 6, 571-581.
Keyanpour, Mohammad, and Azizsefat, Mojtaba (2011). Numerical solution of optimal control
problems by an iterative scheme, AMO- Advanced Modeling and Optimization, Vol. 13,
No. 1, 25-37.
Keyanpour, Mohammad, and Akbarian, Tahereh (2011). Optimal Control of State-Delay
Systems via Hybrid of Perturbation and Parametrization Method, Jour. Advanc. Research.
Dynamic. Cont. Syst, Vol. 3, No. 1, 45-58.
Lotfi, Amir, Dehghanb, Mehdi, and Yousefi, Sohrab Ali (2011). A numerical technique for
solving fractional optimal control problems,Computers and Mathematics with
Applications,article in press doi:10.1016/j.camwa.2011.03.044.
Lakshmikantham, V. and Mohapatra RN. (2001). Strict stability of differen tial equations,
Nonlinear Anal, Vol. 46, No. 7, 915–921.
ሷ
ܱzdemir, Necati, Agrawal, Om Prakash, ‫ ܫ‬sሶ kender, Beyza Billur, and Karadeniz, Derya (2009).
Fractional optimal control of a 2-dimensional distributed system using eigenfunctions,
Nonlinear Dynam, Vol. 55, No. 3, 251–260.
ܱሷzdemir, Necati, Agrawal, Om Prakash, Karadeniz, Derya, and ‫ ܫ‬sሶ kender, Beyza Billur (2009).
Fractional optimal control of an axis-symmetric diffusion-wave propagation. Physica Scripta
T136, Vol. 134, 014024-014029.
ܱሷ zdemir, Necati, Karadeniz, Derya, and ‫ ܫ‬ሶ skender, Beyza Billur (2009). Fractional optimal
control problem of a distributed system in cylindrical coordinates, Physics Letters A, Vol.
373, Issue: 2, 221-226.
Podlubny, Igor (1999). Fractional differential equations. Academic, New York.
Rudin, Walter (1976). Principles of mathematical analysis, 3rd edn, McGraw-Hill.
Rapaic, Milan R and Jelicic, Zoran D. (2010). Optimal Control of a Class of Fractional Heat
Diffusion, Nonlinear Dynam, Vol. 62(1-2), 39-51.
Samko, Stefan, Kilbas, Anatolii Aleksandrovich, Marichev, and Oleg Igorevic (1993). Fractional
integrals and derivatives: Theory and applications, Gordon and Breach, 1006 p, ISBN
2881248640.
Shaw, Michael D. and Yakar, Coşkun (1999). Generalized variation of parameters with initial
time difference and a comparison result in term Lyapunov-like functions, Int J Non-linear
Diff Equations Theory Methods Appl, Vol. 5, 86–108.
Tricaud, Christophe, and Chen, Quan Yang (2010a). An approximate method for numerically
solving fractional order optimal control problems of general form, Computers and
Mathematics with Applications, Vol. 59, No. 5, 1644–1655.
Tangpong, Xiangqing W., and Agrawal Om Prakash (2009). Fractional optimal control of
continuum systems. ASME Journal of Vibration and Acoustics, Vol. 131, No. 2, 021012.1–
021012.1.
Tricaud, Christophe, and Chen, Quan Yang (2010b). Time-Optimal Control of Systems with
Fractional Dynamics, International Journal of Differential Equations, Article ID 461048 16,
pages.
Yousefi, Sohrab Ali, Lotfi, Amir, and Dehghanb, Mehdi (2011). The use of a Legendre
multiwavelet collocation method for solving the fractional optimal control problems, Journal
of Vibration and Control, Vol. 17, No. 13, 2059–2065.
534
T. Akbarian and M. Keyanpour
Yakar, Coskun, Gucen, Mustafa Bayram, and Cicek, Muhammed (2012). Strict Stability of
Fractional Perturbed Systems in Terms of Two Measures, Springer, Eds. Baleanu,
D.,Machado, J.A.T., Luo, A.C.J. Fractional Dynamics and Control, ISBN 978-1-4614-04569, 159-1571.
Wang, JinRong and Zhou, Yong (2011). A class of fractional evolution equations and optimal
controls, Nonlinear Analysis: Real World Applications, Vol. 12, 262–272.